The generator matrix

 1  0  0  0  1  1  1  X  1 X^2+X  1 X^3+X  1  1 X^3+X^2  1  1  1 X^3+X^2 X^3+X X^2  1  0  1  0  1  X  1  1  1  1 X^3 X^3+X^2+X X^3+X^2+X  1 X^3+X^2+X  1  1  1  1  X X^3 X^3+X^2+X  0 X^2  1  1  1  1  1 X^3  X  1  X X^3+X^2+X  1 X^2  X X^3+X^2+X  1  X  1  1  1  1  0  X X^3+X^2 X^3+X^2+X X^3+X X^2  1 X^3+X^2+X  1 X^3+X^2+X  0  1 X^2+X  1 X^3+X X^3+X^2+X X^3+X  1  1 X^2+X  1  1
 0  1  0  0 X^3 X^3+X^2+1 X^3+X+1  1 X^2 X^2 X^2  1 X^2+X+1 X^2+1  1 X^3+1 X^2  0 X^3+X^2  0  1  1  1 X^3+X+1  1 X^3+X  1 X^3+X^2+1  X X^2+X X^3+X^2+X+1  X X^3  X X^3+X^2+1 X^2 X^2+1 X^3+X^2+X X^3+X^2 X^3+X  1  1 X^3+X^2+X X^3+X^2+X  1 X^3+X^2+X+1 X+1 X^3+1  0 X^3+X^2  1  1 X^2+1 X^3+X  X X^2+X  1  1  1 X^2 X^2+X X^3+X^2 X^3+X+1 X^3+X X+1  1 X^3  0  1 X^3+X^2+X  1 X^3+X^2+X  1 X^3+X+1 X^2+X  1 X+1  1 X^2+1  1  1  1 X^2+X+1 X^3+X^2+X+1  1 X^3+1  X
 0  0  1  0 X^3+X^2 X^3 X^2 X^2  1  1 X^3+X+1 X^3+X+1 X^3+1 X+1 X^2+X+1 X^3+X^2+1 X^2+X+1 X^3+X^2+X  1  1  0 X+1 X^3+1 X^3 X^3+X^2+1 X^2+1 X^3+X X^3+X^2+X X^3 X^2+X+1 X^2+X X^3+X^2  1 X^2+X X^3+X^2+X+1  1  X X^2+X  1 X+1 X^2  X  1  1 X^2+1 X^3+X^2+1 X^2+X+1 X^3+X^2 X^3+1 X^3+X^2+X X^2+X X^3  1  1  0 X+1 X^3+X X^2+1 X^2+X+1  X  1 X^3+X+1 X^3+X^2+X+1 X^2+X X^2+X X^3 X^3  1 X^3+X  1 X^2+X+1 X^3+X^2 X^3+1 X^2  0 X^2+1 X^3+X^2+X X^2+X  0 X^3+1 X^3 X^2+X X^3+X+1 X^3+1 X+1 X^2 X^2+1
 0  0  0  1 X^2+X+1 X^3+X^2+X+1 X^3 X+1 X^3+X+1 X^3+X^2+X+1  0 X^3+X^2+1 X^3+X^2+X X^3+X^2+1 X^2+X X^3+X^2+X+1 X^3+1  X X^3+X^2+X+1 X^3 X+1 X^2+X X^3+X^2+X+1  1 X^3+X  X X^3+X^2 X^3 X^2+1 X^3+X X^3  1 X^2+X  1 X^3+X^2 X^3+1 X^2+1 X^2 X^3+X^2 X^3+1 X^3+1  0  X X^3+X+1  1 X^2+X+1 X^2+X+1 X^3+X^2+1 X^2+1 X^3+X+1 X+1 X^2+X X^2 X^2+X+1  1 X+1 X^2+X  0 X^2 X+1  0 X^3+X X^2 X^3+X^2+X+1 X^3+X^2+1 X^3+1  1  X X^3+X^2+1 X^2+1 X^2 X^3+X^2+X X^3+X^2+X+1 X^2+X  1 X^3+X^2+1 X^3+1 X^3+X+1 X^3 X^2+1 X^3+X^2+X+1 X^2 X^3+X^2+X  1 X^3+X^2+X+1 X^3+X+1 X^3

generates a code of length 87 over Z2[X]/(X^4) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+608x^80+1926x^81+3142x^82+4454x^83+5954x^84+6278x^85+7383x^86+6972x^87+7536x^88+6154x^89+5226x^90+3986x^91+2648x^92+1718x^93+905x^94+280x^95+195x^96+64x^97+54x^98+36x^99+10x^100+4x^101+2x^102

The gray image is a linear code over GF(2) with n=696, k=16 and d=320.
This code was found by Heurico 1.16 in 56.1 seconds.